Optimal. Leaf size=35 \[ \frac {\sinh ((1-m) x)}{2 (1-m)}+\frac {\sinh ((m+1) x)}{2 (m+1)} \]
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Rubi [A] time = 0.03, antiderivative size = 35, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 2, integrand size = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {5614, 2637} \[ \frac {\sinh ((1-m) x)}{2 (1-m)}+\frac {\sinh ((m+1) x)}{2 (m+1)} \]
Antiderivative was successfully verified.
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Rule 2637
Rule 5614
Rubi steps
\begin {align*} \int \cosh (x) \cosh (m x) \, dx &=\int \left (\frac {1}{2} \cosh ((1-m) x)+\frac {1}{2} \cosh ((1+m) x)\right ) \, dx\\ &=\frac {1}{2} \int \cosh ((1-m) x) \, dx+\frac {1}{2} \int \cosh ((1+m) x) \, dx\\ &=\frac {\sinh ((1-m) x)}{2 (1-m)}+\frac {\sinh ((1+m) x)}{2 (1+m)}\\ \end {align*}
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Mathematica [A] time = 0.04, size = 25, normalized size = 0.71 \[ \frac {m \cosh (x) \sinh (m x)-\sinh (x) \cosh (m x)}{m^2-1} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.42, size = 42, normalized size = 1.20 \[ \frac {m \cosh \relax (x) \sinh \left (m x\right ) - \cosh \left (m x\right ) \sinh \relax (x)}{{\left (m^{2} - 1\right )} \cosh \relax (x)^{2} - {\left (m^{2} - 1\right )} \sinh \relax (x)^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.11, size = 59, normalized size = 1.69 \[ \frac {e^{\left (m x + x\right )}}{4 \, {\left (m + 1\right )}} + \frac {e^{\left (m x - x\right )}}{4 \, {\left (m - 1\right )}} - \frac {e^{\left (-m x + x\right )}}{4 \, {\left (m - 1\right )}} - \frac {e^{\left (-m x - x\right )}}{4 \, {\left (m + 1\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.18, size = 28, normalized size = 0.80 \[ \frac {\sinh \left (\left (-1+m \right ) x \right )}{-2+2 m}+\frac {\sinh \left (\left (1+m \right ) x \right )}{2+2 m} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: ValueError} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.07, size = 26, normalized size = 0.74 \[ -\frac {\mathrm {cosh}\left (m\,x\right )\,\mathrm {sinh}\relax (x)-m\,\mathrm {sinh}\left (m\,x\right )\,\mathrm {cosh}\relax (x)}{m^2-1} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.92, size = 56, normalized size = 1.60 \[ \begin {cases} - \frac {x \sinh ^{2}{\relax (x )}}{2} + \frac {x \cosh ^{2}{\relax (x )}}{2} + \frac {\sinh {\relax (x )} \cosh {\relax (x )}}{2} & \text {for}\: m = -1 \vee m = 1 \\\frac {m \sinh {\left (m x \right )} \cosh {\relax (x )}}{m^{2} - 1} - \frac {\sinh {\relax (x )} \cosh {\left (m x \right )}}{m^{2} - 1} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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